John E. Opfer

Representational change and children's numerical estimation

We applied overlapping waves theory and microgenetic methods to examine how children improve their estimation proficiency, and in particular how they shift from reliance on immature to mature representations of numerical magnitude. We also tested the theoretical prediction that feedback on problems on which the discrepancy between two representations is greatest will cause the greatest representational change. Second graders who initially were assessed as relying on an immature representation were presented feedback that varied in degree of discrepancy between the predictions of the mature and immature representations. The most discrepant feedback produced the greatest representational change. The change was strikingly abrupt, often occurring after a single feedback trial, and impressively broad, affecting estimates over the entire range of numbers from 0 to 1000. The findings indicated that cognitive change can occur at the level of an entire representation, rather than always involving a sequence of local repairs.

Revisiting Preschoolers' Living Things Concept: A Microgenetic Analysis of Conceptual Change in Basic Biology.

Many preschoolers know that plants and animals share basic biological properties, but this knowledge does not usually lead them to conclude that plants, like animals, are living things. To resolve this seeming paradox, we hypothesized that preschoolers largely base their judgments of life status on a biological property, capacity for teleological action, but that few preschoolers realize that plants possess this capacity. To test the hypothesis, we taught 5-year-olds one of four biological facts and examined the childrens subsequent categorization of life status for numerous animals, plants, and artifacts. As predicted, a large majority of 5-year-olds who learned that both plants and animals, but not artifacts, move in goal-directed ways inferred that both plants and animals, but not artifacts, are alive. These children were considerably more likely to draw this inference than peers who learned that the same plants and animals grow or need water and almost as likely to do so as children who were explicitly told that animals and plants are living things and that artifacts are not. Results also indicated that not all biological properties are extended from familiar animals to plants; some biological properties are first attributed to plants and then extended to animals.

How 15 hundred is like 15 cherries: effect of progressive alignment on representational changes in numerical cognition.

How does understanding the decimal system change with age and experience? Second, third, sixth graders, and adults (Experiment 1: N = 96, mean ages = 7.9, 9.23, 12.06, and 19.96 years, respectively) made number line estimates across 3 scales (0-1,000, 0-10,000, and 0-100,000). Generation of linear estimates increased with age but decreased with numerical scale. Therefore, the authors hypothesized highlighting commonalities between small and large scales (15:100::1500:10000) might prompt children to generalize their linear representations to ever-larger scales. Experiment 2 assigned second graders (N = 46, mean age = 7.78 years) to experimental groups differing in how commonalities of small and large numerical scales were highlighted. Only children experiencing progressive alignment of small and large scales successfully produced linear estimates on increasingly larger scales, suggesting analogies between numeric scales elicit broad generalization of linear representations.

Costs and benefits of representational change: Effects of context on age and sex differences in symbolic magnitude estimation

Studies have reported high correlations in accuracy across estimation contexts, robust transfer of estimation training to novel numerical contexts, and adults drawing mistaken analogies between numerical and fractional values. We hypothesized that these disparate findings may reflect the benefits and costs of learning linear representations of numerical magnitude. Specifically, children learn that their default logarithmic representations are inappropriate for many numerical tasks, leading them to adopt more appropriate linear representations despite linear representations being inappropriate for estimating fractional magnitude. In Experiment 1, this hypothesis accurately predicted a developmental shift from logarithmic to linear estimates of numerical magnitude and a negative correlation between accuracy of numerical and fractional magnitude estimates (r=-.80). In Experiment 2, training that improved numerical estimates also led to poorer fractional magnitude estimates. Finally, both before and after training that eliminated age differences in estimation accuracy, complementary sex differences were observed across the two estimation contexts.

Children are not like older adults: A diffusion model analysis of developmental changes in speeded responses

Children (n = 130; M(age) = 8.51-15.68 years) and college-aged adults (n = 72; M(age) = 20.50 years) completed numerosity discrimination and lexical decision tasks. Children produced longer response times (RTs) than adults. R. Ratcliffs (1978) diffusion model, which divides processing into components (e.g., quality of evidence, decision criteria settings, nondecision time), was fit to the accuracy and RT distribution data. Differences in all components were responsible for slowing in children in these tasks. Children extract lower quality evidence than college-aged adults, unlike older adults who extract a similar quality of evidence as college-aged adults. Thus, processing components responsible for changes in RTs at the beginning of the life span are somewhat different from those responsible for changes occurring with healthy aging.

How Numbers Bias Preschoolers’ Spatial Search

Numbers often bias adults’ spatial performance. Because the direction of this bias (left-to-right versus right-to-left) is culture-specific, it has been assumed that spatial-numeric associations develop with reading practice or schooling. The authors tested this assumption by examining spatial-numeric associations in pre-reading preschoolers. Preschoolers were shown two boxes (sample and matching boxes) subdivided into seven verbally numbered “rooms” (e.g., “the four room”). A “winner” card was revealed in the sample box, and children searched for the “winner” in the matching box (located in the same-numbered room). Preschoolers were faster and more accurate when rooms increased numerically from left-to-right versus right-to-left. This advantage was apparently caused by numbers influencing preschoolers’ encoding of spatial locations: Ordering of numbers in the sample box affected preschoolers’ search greatly, whereas ordering of numbers in the matching box did not. The authors conclude that numeric effects on spatial encoding develop far too early to be caused by reading practice or schooling.

Cognitive Constraints on How Economic Rewards Affect Cooperation

Cooperation often fails to spread in proportion to its potential benefits. This phenomenon is captured by prisoners dilemma games, in which cooperation rates appear to be determined by the distinctive structure of economic incentives (e.g.,

Development of Spatial-Numerical Associations

3 for mutual cooperation vs.

Representational change and magnitude estimation: Why young children can make more accurate salary comparisons than adults☆

5 for unilateral defection). Rather than comparing economic values of cooperating versus not (

Free versus anchored numerical estimation: A unified approach

3 vs.